Vectors MCQ Quiz - Objective Question with Answer for Vectors - Download Free PDF

Calculation:

The x-component of OA = (OA) cos 90° = 0

The x-component of OB = (OB) cos 0° = OB

The x-component of OC = (OC) cos 135° = – (1/√2) OC

Hence, the x-component of the resultant:

⇒ OB – (1/√2) OC    ... (i)

It is given that the resultant is zero, and hence its x-component is also zero. From (i):

⇒ OB = (1/√2) OC     ... (ii)

The y-component of OA = OA cos 180° = – OA

The y-component of OB = OB cos 90° = 0

The y-component of OC = (OC) cos 45° = (1/√2) OC

Hence, the y-component of the resultant:

⇒ (1/√2) OC – OA     ... (iii)

As the resultant is zero, so is its y-component. From (iii):

(1/√2) OC = OA

⇒ OC = √2 OA = 5√2 m

From (ii), OB = (1/√2) OC = 5 m

Thus the OB / OC = 1/ √2 = sin(π/4) 

Calculation:

We can easily add two or more vectors if we know their components along the rectangular coordinate axes. Let us have:

a = a_x î + a_y ĵ + a_z k̂

b = b_x î + b_y ĵ + b_z k̂

c = c_x î + c_y ĵ + c_z k̂

Then, a + b + c = (a_x + b_x + c_x) î + (a_y + b_y + c_y) ĵ + (a_z + b_z + c_z) k̂

If all the vectors are in the X-Y plane, then all the z components are zero, and the resultant is simply:

a + b + c = (a_x + b_x + c_x) î + (a_y + b_y + c_y) ĵ

This is the sum of two mutually perpendicular vectors of magnitude (a_x + b_x + c_x) and (a_y + b_y + c_y).

The resultant can easily be found to have a magnitude: √[(a_x + b_x + c_x)² + (a_y + b_y + c_y)²]

making an angle α with the X-axis where: tan α = (a_y + b_y + c_y) / (a_x + b_x + c_x)

Thus, The angle made is tanα = 4/3 ⇒ α = 53^∘ .

Calculation:

The magnitude of the resultant vector:

R' = √(x² + y² + 2xy cos θ)

Here, x = y = a

Then, R' = √(a² + a² + 2a² cos θ)

⇒ R' = a√2 √(1 + cos π ) = 0

∴ The correct answer is Option (1)

Calculation:

The point A is (a, 0, a) and point B is (0, a, a).

The vector AF will be:

AF = a î − a ĵ

The unit vector will be:

n̂ = (1 / √2) (î − ĵ)

Concept:

Let the magnitude of (A + B) be R and the magnitude of (A − B) be R'.

Given, the magnitude of R is √3 times the magnitude of R'.

Thus, R = √3 R'

Now,

R = A + B

R² = A² + B² + 2AB cos θ

⇒ R² = A² + A² + 2AA cos θ

R² = 2A² + 2A² cos θ

Again,

R' = A − B

R'² = A² + B² − 2AB cos θ

⇒ R'² = A² + A² − 2AA cos θ

R'² = 2A² − 2A² cos θ

(R / R')² = 3

3 = (1 + cos θ) / (1 − cos θ)

⇒ 4 cos θ = 2

⇒ θ = cos⁻¹(1 / 2)

CONCEPT:

• All measurable quantities are divided into two broad categories:

EXPLANATION:

• Energy is quantity has only magnitude. It does not require any direction. So it is a scaler, not vector quantity.
• The weight of a body denotes gravitational force. And force requires magnitude and direction (in which direction it is acting) both to describe.
  • So weight is a vector.
• Similarily momentum requires magnitude and direction (in which direction it is acting) both to describe.
  • So momentum is a vector.
• In chemistry, you will study that the Nuclear spin of an electron has two directions clockwise and anti-clockwise. So it will also be a vector. 

CONCEPT:

• Speed: The rate of change of distance is called speed.
  • It is a scalar quantity.
• Velocity: The rate of change of displacement is called velocity. 
  • It is a vector quantity.
• Scalar quantities: The physical quantities which have only magnitude and no direction are called scalar quantities or scalars.
  • Examples: Mass, volume, density, time, temperature, electric current, Luminious intensity, speed, etc.
• Vector quantities: The physical quantities which have both magnitude and direction and obey the laws of vector addition are called vector quantities or vectors.
  • Examples Displacement, velocity, acceleration, force, momentum, Impulse, etc.

EXPLANATION:

1. Time is a scalar quantity.
2. Volume is a scalar quantity.
3. Speed is a scalar quantity.
4. Velocity is a vector quantity. So option 4 is correct.

CONCEPT:

• All measurable quantities are divided into two broad categories:

EXPLANATION:

• From the above table, it is clear that force, velocity, and acceleration are vector quantity because they have both magnitude as well as direction. Therefore option 1, 2, and 3 is incorrect.
• Pressure is defined as force per unit area. It is a scalar quantity because it has only magnitude and it is independent on the size of the area chosen. Therefore option 4 is correct.

The correct answer is 40N

CONCEPT:​

• Resolution of vectors into components: We have a vector (F) where the magnitude of the vector is F and the angle with horizontal is θ.

The vector has two components: 1. Vertical component and 2. Horizontal component

Vertical component (F_y) = F Sinθ

Horizontal component (F_x) = F Cosθ 

Here \(F = \sqrt {F_x^2 + F_y^2}\)

CALCULATION:

Here F₁ and F₂ are along X- and Y- direction.

Let the applied force F = 50

And the x-component of the applied force F_x = 30

The y-component of the applied force F_y = ?

We know that the vector sum of the force

\(F = \sqrt {F_x^2 + F_y^2}\)

\(50N = \sqrt {{{30}^2} + {F^2}}\)

Now squaring both sides

2500 = 900 + F²

\({F_y} = \sqrt {2500 - 900} = \sqrt {1600}\)

\({F_y} = 40N\)

So option 3 is correct.

CONCEPT:

• Speed: The rate of change in distance is called speed. 
  • It is a scalar quantity. It is the magnitude of velocity that can never be negative.
• Mass: The quantity of matter in any object is called the mass of that object.
  • It can never be negative. It is a scalar quantity.
• Velocity: The rate of change in displacement is called velocity.
  • It is a vector quantity that can be negative, positive, or zero.
• Distance: The total path length between two points is called distance.
  • It is a scalar quantity and can never be negative.

EXPLANATION:

• As discussed above, velocity is a vector quantity that can be negative. So option 3 is correct.

Additional Information

• Scalar quantities: The physical quantities which have only magnitude and no direction are called scalar quantities or scalars.
  • A scalar quantity can be specified by a single number, along with the proper unit.
  • Examples: Mass, volume, density, time, temperature, electric current, Luminious intensity, etc.
• Vector quantities: The physical quantities which have both magnitude and direction and obey the laws of vector addition are called vector quantities or vectors.
  • A vector quantity is specified by a number with a unit and its direction.
  • Examples Displacement, velocity, force, momentum, etc

CONCEPT:

• All measurable quantities are divided into two broad categories:

EXPLANATION:

• From the above, it is clear that mass, length, and speed are scalar quantity because they have the only magnitude. Therefore option 1, 2, and 3 is incorrect.

• Impulse is a vector quantity because the force is a vector quantity. So option 4 is an example of a vector quantity.

CONCEPT:

• Acceleration (a): The rate of change of the velocity of an object is called acceleration.
  • It is a vector quantity.
• Scalar quantities: The physical quantities which have only magnitude and no direction are called scalar quantities or scalars.
  • Examples: Mass, volume, density, time, temperature, electric current, Luminious intensity, etc.
• Vector quantities: The physical quantities which have both magnitude and direction and obey the laws of vector addition are called vector quantities or vectors.
  • Examples Displacement, velocity, acceleration, force, momentum, Impulse, etc.

EXPLANATION:

• Acceleration is a vector quantity. So option 1 is correct.

CONCEPT:

• Force: The interaction which after applying on a body changes or try to change the state of rest or state of motion is called force.
  • It is a vector quantity.
• Temperature: The measurement of hotness is called temperature.
  • It is a scalar quantity.
• Scalar quantities: The physical quantities which have only magnitude and no direction are called scalar quantities or scalars.
  • Examples: Mass, volume, density, time, temperature, electric current, Luminious intensity, etc.
• Vector quantities: The physical quantities which have both magnitude and direction and obey the laws of vector addition are called vector quantities or vectors.
  • Examples Displacement, velocity, acceleration, force, momentum, Impulse, etc.

EXPLANATION:

1. Displacement is a vector quantity as it is directed from the initial point to the final point. 
2. Current: The rate of flow of electric charge is called current.
3. Temperature is NOT a vector quantity. It is a scalar quantity. So option 2 is correct.
4. Drag is also a type of force and it is a vector quantity.
5. Force is a vector quantity.

CONCEPT:

The dot product of vector:

• The dot product is the sum of the products of the corresponding entries of the two sequences of numbers.
• Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them.

\({{\rm{A}}_1}\cdot{{\rm{A}}_2} = \left| {\overrightarrow {{{\rm{A}}_1}} } \right|\left| {\overrightarrow {{{\rm{A}}_2}} } \right|\cos {\rm{θ }}\)

Where \(\left| {\overrightarrow {{{\rm{A}}_1}} } \right|.\left| {\overrightarrow {{{\rm{A}}_2}} } \right|\) are the magnitudes of two vectors A1 and A2

EXPLANATION:

• The angle between two vectors \(\vec A\) and \(\vec B\) given by

\(\Rightarrow cos\;\theta = \frac{{\vec A.\vec B}}{{\left| {\vec A} \right|\left| {\vec B} \right|}}\)

• Therefore option 4 is correct.

Concept:

Physical quantities are of two types:

Explanation:

Electric Field Intensity (E):

• The space around an electric charge in which its influence can be felt is known as the electric field.
• The electric field intensity at a point is the force experienced by a unit positive charge placed at that point.
• Electric Field Intensity is a vector quantity. It is denoted by ‘E’.
• Electric Field = F/q.
• Unit of E is NC-1 or Vm-1

Electrostatic energy and electrostatic potential are scalar quantities because it only requires magnitude and not the direction.

Mistake Points

• In the case of electric current, when two currents meet at a junction, the resultant current of these will be an algebraic sum and not the vector sum.
• Therefore, an electric current is a scalar quantity although it possesses magnitude and direction.